256 ANALYTICAL MECHANICS where r is the density of water. Substituting this expression for m in equation (1) 0 (a + «)«=! [(a + «)»«] at = (a + kt)*^ + Z(a-rkt)*kv at , dv , 3k * ,~ and it + ^+ktv = g> (2>' r s k r 3 k the integral of which is v = e a+kt | Cge a + kt dt + c | • /. v = e~31°£ <fl + *3 [JV10* (a + *<> & 4- cl = (a + ta)~3 [V J(a + &03 d« + cl / .7^ .T ( v , 3a2^2 , 3afe2^3 , ^4\ , I = (a + ^)-8^ (^ + — — + -y- + — J + cj = (a + ^)-3f| (4 a3^ + 6 a%2 + 4 akH* + kH*) + c] . Let # = 0 when t — 0; then c = 0. fi? 4 a3 + 6 a2Jc^ + 4 a/b2^2 + kH* PROBLEMS. 1. Find the pressure upon the canvas roof of a tent produced by a shower. The following data are given — the raindrops have a velocity of 50 — '- at right angles to the roof; the intensity of the shower is such as sec. to produce a deposit of 0.2 inch per hour; 1 cubic foot of water weighs 62.5 pounds. 2. Find the pressure on horizontal ground due to the impact of a column of water which falls vertically from a height of 500 feet. cm 3. Water flowing through a pipe at the rate of 100 — '• is brought to- sec. * Equation (2) is of the form ~t- -{- Py = Q, which is the typical linear equation, with the integral y = e~~*'p<2aT CQe^^dx + cl.